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The Jahn - Teller (JT) problem is investigated analytically using a unitary transformation method. Minimization of the adiabatic energy surface for this problem results in wells of either or symmetry, depending on the coupling strengths. The dynamic JT problem is then solved in the tunnelling regime using projection operators to find symmetrized combinations of the states associated with the wells. By analogy to other JT systems, the ground state would be expected to have the same degeneracy as the original orbital state, and thus to be an H-type quintet. However, it is found that there are a range of couplings strengths for the g and h modes for which the tunnelling ground state for the wells can be an A-type singlet. A similar result was recently found for the pure JT system. It is also found that for wells, the limiting value of the tunnelling splitting between the H and A states for a pure system tends to in weak coupling, whilst for a pure system it tends to . For systems coupled to both modes, the value of the tunnelling splitting strongly depends upon which of the two modes is dominant. Both the level ordering in strong coupling and the anomalous behaviour in weak coupling can be shown to be fundamental symmetry properties of these JT systems, and not consequences of the details of our model. The JT systems studied here are possible models for the ground state of the cation and for an excited state of the anion .

First- and second-order vibronic reduction factors are calculated analytically for the H⊗(g⊕h) Jahn-Teller system in Ih symmetry. Results are given as a function of the strength of the coupling of the H orbital to the vibrations of h and g symmetries. As the product systems H⊗g and H⊗h contain repeated representations, the calculations of many of the reduction factors are more complicated than in other systems. These complications and their implications are analysed in detail. This system models the ground state of hole-doped C60 material, which has possible applications for high-temperature superconductivity.

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Jahn–Teller (JT) coupling between electronic motion and lattice or molecular vibrations results in an adiabatic potential energy surface that contains either wells or troughs of minimum-energy points. When wells are lowest in energy, the system will vibrate about the minimum-energy points. This vibration must be taken into account when describing the quantum mechanical states of the system. In general, the wells will be intrinsically anisotropic. This anisotropy alters the vibrational frequencies and hence the positions of the energy levels, and can be particularly significant when the barriers between wells are shallow. In this paper, we will show how anisotropic states and their energies can be calculated using two unitary transformations. The first locates minima on the adiabatic potential energy surface, and the second accounts for anisotropy in the shape of the minima. The method is developed in a way general enough to allow it to be applied to any linear JT problem. The theory is then applied to the icosahedral H ⊗ (h ⊕ g) JT system. The results obtained will help the understanding of, for example, the effects of vibronic coupling in positively charged fullerene ions.